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Section 9.3 Distribution of Sample Means

Section 9.3 Distribution of Sample Means. AP Statistics February 5, 2010 Berkley High School. Sampling from a Normal Distribution. If you sample from a population that is normal, the resulting distribution of sample means is also normal.

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Section 9.3 Distribution of Sample Means

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  1. Section 9.3Distribution of Sample Means AP Statistics February 5, 2010 Berkley High School

  2. Sampling from a Normal Distribution • If you sample from a population that is normal, the resulting distribution of sample means is also normal. • If the population distribution is N(μ,σ), the distribution of sample means is N(μ,σ/n^.5) • Restrictions: population size at least 10 times sample size. AP Statistics, Section 9.3

  3. Central Limit Theorem • Draw an SRS of size n from any population whatsoever with mean μ and finite standard deviation σ. When n is large, the sampling distribution of the sample mean x-bar is close to the normal distribution N(μ,σ/n^.5), with mean μ and standard deviation σ/n^.5. AP Statistics, Section 9.3

  4. AP Statistics, Section 9.3

  5. Example • The time that a technician requires to perform preventive maintenance on an air-conditioning unit is governed by the exponential distribution whose density curve has mean 1 hour and standard deviation 1 hour. • Your company operates 70 of these units. • What is the probability that their average maintenance exceeds 50 minutes? AP Statistics, Section 9.3

  6. Example: • Are the restrictions of sampling fulfilled? • Sampling size of 70 is less than one tenth of the infinite number of the “maintenances” that could be done. • The population distribution is not normal, but the sample of 70 is large enough for the distribution of sample means to be approximately normal with mean 1 and standard deviation 1/70^.5. AP Statistics, Section 9.3

  7. AP Statistics, Section 9.3

  8. AP Statistics, Section 9.3

  9. Exercises • 9.35-9.41 all AP Statistics, Section 9.3

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